Question

1. The radius of a regular decagon is 6m. What is the length of its apothem?
2. The area of a triangle with sides of length 10 in. and 17 in. and an included angle of 113° is equal to the area of a regular heptagon. Determine the length of each side of the heptagon.
3. Determine the area of the waste material in cutting out the largest circle (diameter is 23 cm) from a regular decagon.

Answer 1

Answer on Question #57221 – Math – Geometry

1. The radius of a regular decagon is $6\mathrm{m}$. What is the length of its apothem?

Solution

Definition: Apothem is a line segment from the center of a regular polygon to the midpoint of a side.

If you know the radius (distance from the center to a vertex), then the length of apothem is given by

$$apothem = r \cos \left(\frac {180}{n}\right),$$

where

$r$ is the radius of the polygon,

$n$ is the number of sides,

$\cos$ is the cosine function calculated in degrees.

$$apothem = 6 \cos \frac {180}{10} = 6 \cos 18 = 5.71\,\mathrm{m}.$$

Answer: $5.71\,\mathrm{m}$.

2. The area of a triangle with sides of length 10 in. and 17 in. and an included angle of $113{}^\circ$ is equal to the area of a regular heptagon. Determine the length of each side of the heptagon.

Solution

$$A_{\text{triangle}} = \frac{1}{2} 10 \cdot 17 \sin 113{}^\circ = 75.74$$

By definition, all sides of a regular polygon are equal in length.

If you know the length of one of the sides, then the area is given by the formula:

$$area = \frac{s^2 N}{4 \tan \left(\frac{180}{N}\right)}$$

where $s$ is the length of any side,

$N$ is the number of sides,

$\tan$ is the tangent function calculated in degrees.

Given the area of a regular heptagon is

$$A_{\text{heptagon}} = A_{\text{triangle}} = 75.74.$$

On the other hand, by the formula (1)

$$A_{\text{heptagon}} = \frac{7s^2}{4 \tan \left(\frac{180}{7}\right)} \Rightarrow s = \sqrt{\frac{4}{7} A_{\text{heptagon}} \tan \left(\frac{180}{7}\right)} = \sqrt{\frac{4}{7} \cdot 75.74 \cdot 0.48} = 20.77\,\mathrm{in}.$$

$s = 20.77\,\mathrm{in}$, where $s$ is the length of each side of the heptagon.

Answer: 20.77 in.

3. Determine the area of the waste material in cutting out the largest circle (diameter is 23 cm) from a regular decagon.

Solution

The area of the waste material is

$$A_{\text{waste material}} = A_{\text{decagon}} - A_{\text{circle}}$$

The area of the circle is

$$A_{\text{circle}} = \left(\frac{23}{2}\right)^2 \pi = 415.27$$

If you know the apothem, or inradius, then the area of a regular polygon is given by

$$\text{area} = A^2 N \tan \left(\frac{180}{N}\right)$$

where

$A$ is the length of the apothem (inradius),

$N$ is the number of sides,

$\tan$ is the tangent function calculated in degrees.

The area of the regular decagon is

$$A_{\text{decagon}} = \left(\frac{23}{2}\right)^2 \cdot 10 \cdot \tan 18 = 429.71$$

Then

$$A_{\text{waste material}} = 429.71 - 415.27 = 14.44 \, \text{cm}^2.$$

Answer: $14.44 \, \text{cm}^2$.

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